Non-Hermitian systems have become a multifaceted research frontier due to their substantial number of applications in diverse emerging fields [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. This theoretical formulation, an extension of conventional quantum mechanics in the complex domain, has opened an alternative way of constructing strange-looking Hamiltonians with real spectra without the stringent Hermiticity requirement [8]}, [9]}, [10]}, [11]}. In a series of papers, Bender and colleagues [8]}, [9]}, [10]}, [15]}, [16]}, [17]} proved that the Hermiticity condition can be relaxed under the assumption of \({\mathcal {P}T}\)  symmetry; naturally, Bender's pioneering work has marked the grounds for these non-Hermitian Hamiltonians to acquire a new meaning with \({\mathcal {P}T}\) -symmetric quantum mechanics [18]}, [19]}, [20]}, [21]}. These concepts indeed have already led to compelling applications in optics; for instance, \({\mathcal {P}T}\) -symmetric Hamiltonians found an ideal paragon on coupled waveguides differentiated by absorption and amplification into their refractive index profiles [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}. On the other hand, in the light of elementary quantum theory, the acceptance of complex Hamiltonians has been flourishing for a new framework where \({\mathcal {P}T}\)  symmetric Hamiltonians can be mapped into Hermitian ones via non-unitary transformations [29]}, [30]}, [31]}. Following the seminal insight of these developments, we hence proceed in this work to look for non-unitary transformations that applied to Hermitian Hamiltonians, i.e. Hamiltonians that have real eigenvalues, produce \(\mathcal {PT}\) -symmetric Hamiltonians, and vice versa. We then show some specific transformations that produce some renowned \(\mathcal {PT}\) -symmetric Hamiltonians; in particular, we show also that the non-unitary transformation may be associated, via a fractional-Wick rotation, to complex time.
